The Food and Drug Administration (“FDA”) sets guidelines for testing methods utilized in the pharmaceutical and related industries. The FDA evaluates whether a particular analytical method is suitable for its intended purpose. Once it is established that the method is suitable, the method is “validated.”
In order to validate a testing method, the FDA requires an applicant to evaluate many differing characteristics of the method. Although not all the characteristics of a particular method must be demonstrated in each case, the linearity of the relationship between an actual analyte concentration and a test result from the method is required for all quantitative methods.
Linearity is independent of the technology used to ascertain the analyte concentration. For instance, even the most modern instrumental methods that rely on multivariate chemometric computer methods have to produce a number that represents a final answer for the analyte, which would be the test result from the instrument. Therefore, the term “linearity” applies to all types of analytical methodology from manual wet chemistry to the latest high-tech instrument.
The FDA guidelines provide various definitions of the meaning of the term “linearity”. For instance, one definition is: “ . . . ability (within a given range) to obtain test results which are directly proportional to the concentration (amount) of analyte in the sample.” This is a definition that is essentially unattainable in practice when noise and error are taken into account. For instance, a set of hypothetical data points that most would agree represents a substantially linear relationship between a test result and an analyte concentration may be plotted. However, even though there is a line that meets the criterion that “test results are directly proportional to the concentration of analyte in the sample”, none of the data points may actually fall on the line. Therefore, based upon the FDA definition, none of the data points representing the test results can be said to be proportional to the analyte concentration.
Differing descriptions of linearity are also provided. For instance, one recommendation is visual examination of a plot (unspecified, but presumably also of the method response versus the analyte concentration). Because this method requires a visual examination, it is inherently subjective and not amenable to the application of statistical tests, making an objective mathematical evaluation unattainable. This method is also open to different interpretations, and is unsuitable for application with computerized or automated screening methods.
A further recommendation in the guidelines is to use “statistical methods”; where calculation of a linear regression line is advised. This however, is not so much a definition of linearity, as an attempt to evaluate linearity. For instance, if regression is performed, then the correlation coefficient, slope, y-intercept and residual sum of squares are determined. However, there are no guidelines as to how these quantities are to be related to linearity. One reference by F. J. Anscombe, Amer. Stat. 27 pp. 17-21, presents several (synthetic) data sets, which are fit to a straight line using Least Squares regression. One data set is substantially linear, while another is a data set that is non-linear. However, when linear regression is performed on any of these data sets as recommended by the guidelines, all the recommended regression statistics are identical for the sets of data. It is immediately observed that the linear regression results cannot distinguish between the two cases, since the regression results are the same for both of them.
Other linearity tests exist, in addition to the ones in official guidelines. One such proposed test is the Durbin-Watson (“DW”) statistic, for use as a statistically based test method for evaluating linearity. However, it has been determined that use of the DW statistic provides unsatisfactory results. For instance, DW for residuals from regression data that are random, independent, normally distributed and represent a linear relation between two variables has an expected value of two. (See Draper, N., Smith, H., “Applied Regression Analysis” 3 ed., John Wiley & Sons, New York (1998) pp. 180-185). However, a fatal flaw in the DW method for use in this regard may be shown by calculating the DW statistic for the data sequence: 0, 1, 0, −1, 0, 1, 0, −1, 0, 1, 0, −1, . . . which, also results in a computed value of two, despite the fact that this sequence is non-random, non-independent, not normally distributed and not linear. Sets of residuals showing a similar cyclic behavior also compute out to a value of DW that will erroneously indicate satisfactory behavior of the residuals.
Another test is a statistical F-test. An F-test is based on comparing sample estimates to the overall error of the analysis. This test is undesirable because it is insensitive. For instance, any bias in the estimates of the concentration will inflate the F-value, which will be taken as an indicator of non-linearity when some other phenomenon may be affecting the data. Furthermore it requires multiple readings of every sample by both the method under test and the method used to determine the actual concentration of the analyte, making it impractical to apply on a routine basis, and inapplicable to already existing data.
Still another method is disclosed by Hald, A., “Statistical Theory with Engineering Applications”, John Wiley & Sons, Inc. New York (1952). Hald recommends testing whether the residuals are normally distributed since it is unlikely that the residuals will be normally distributed if there is appreciable non-linearity in the relationship between concentration and the test results. However, this test is again insensitive to actual non-linearity (especially for small numbers of samples), and also suffers from the same difficulties as the F-test, namely that other types of problems with the data may be erroneously called non-linearity.
None of the above-mentioned methods are completely satisfactory for utilization in the pharmaceutical and related industries. In fact, the recommendations of the official guidelines for evaluating linearity, both the definitions and the recommended method(s) for assessing it are themselves not suitable for their intended purpose.
Therefore what is desired is to provide a new method for reliably testing the linearity of data.
It is further desired to provide statistical results that the current FDA test procedure recommends in a context that makes those statistics more meaningful.
It is further desired to provide the derivation and details of the operation for the new method of evaluating data.
It is also desired to disclose a report on the ability of the new method to test linearity by applying it to data from a real analytical method.
It is further desired to disclose a report on the ability of the new method to test linearity of Near Infra-Red (“NIR”) spectroscopic analysis using diffuse transmittance measurements.
It is still further desired to disclose a report on the ability of the new method to test linearity of NIR spectroscopic analysis using diffuse reflectance measurements.